Order regularity of two-node Birkhoff interpolation with lacunary polynomials

Interpolation of Shifted-Lacunary Polynomials [Extended Abstract]

Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 < e2 < · · · < et, and the coefficients c1, . . . , ct ∈ Q \ {0} such that f (x) = c1(x − α)1 + c2(...

Birkhoff Interpolation with Rectangular Sets of Nodes

Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Here we explain the problem, and then we concentrate on one of the simplest shapes (“rectangular”) in the case of bivariate uniforme Birkhoff interpolation schemes. The ultimate goal is to obtain a geometrical understanding ...

On Multivariate Birkhoff Rational Interpolation

Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights

* Correspondence: hsun90@skku. edu Department of Mathematics Education, Sungkyunkwan University Seoul 110-745, Republic of Korea Full list of author information is available at the end of the article Abstract Let R = [0, ∞) and R : R ® R be a continuous function which is the Laguerretype exponent, and pn, r(x), ρ > − 2 be the orthonormal polynomials with the weight wr(x) = x r e. For the zeros ...

Lower Bounds by Birkhoff Interpolation

In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order Ω( √...